## Zhongda Tian* and Chao Zhang*## |

Functions | Dimensions | Range of parameters | Optimal value | Permissible error |

f _{ 1 } | 30 | [-5.12, 5.12] | 0 | 15 |

f _{ 2 } | 30 | [-500, 500] | 0 | 10 |

f _{ 3 } | 30 | [-30, 30] | 0 | 4 |

f _{ 4 } | 30 | [-2.048, 2.048] | 0 | 10 |

In order to eliminate the effects of randomness, all algorithms run 20 times. Selecting the average value as a result optimization, the fitness function testing processes of four convergence curves are shown in Figs. 4–7. 5000 is considered as the number of iterations, for ease of illustration, the horizontal ordinate adaptation values recorded every 50 iterations. Thus the level of the ordinate is a function of the range of 0 to 100. The value for the optimization function, the degree of adaptation is different values of x. Since the range of variation (fitness) function value is too large; longitudinal coordinate of the use forms 10. The optimization problem in this paper is to find the global minimum of the test function through the harmony search algorithm, so as the number of iterations increases, the curves in Figs. 3 and 4 are also the same. Figs 4–7 are monotonically decreasing. When the value of the function to find the global optimum adaptation curve remains unchanged. It should be noted that some random factors present optimization process, so there are some differences in the shape of the curve of each optimization. As can be seen from these figures, the IHS algorithm is better than the standard algorithm, the IHS algorithm and the EHS algorithm converges faster and have better fitness.

Table 2.

Function | Algorithm | Average fitness | Best average | Standard deviation | Success rate (%) |

f _{ 1 } | HS | 13.6480 | 11.3287 | 2.0176 | 55 |

IHS | 11.2023 | 10.5874 | 1.5621 | 60 | |

EHS | 8.4558. | 6.7412 | 1.2354 | 65 | |

Algorithm in this paper | 0.2939 | -0.2435 | 0.0623 | 100 | |

f _{ 2 } | HS | 10.0691 | 7.5561 | 1.7824 | 50 |

IHS | 9.2632 | 7.1236 | 1.5202 | 65 | |

EHS | 7.2102 | 5.3214 | 1.0232 | 70 | |

Algorithm in this paper | 0.0748 | 0.0008 | 0.0695 | 100 | |

f _{ 3 } | HS | 6.2356 | 3.1523 | 2.4232 | 45 |

IHS | 5.2541 | 2.321 | 1.9523 | 65 | |

EHS | 2.2023 | 0.9523 | 1.1021 | 80 | |

Algorithm in this paper | 0.6089 | 0.3598 | 0.0063 | 100 | |

f _{ 4 } | HS | 15.1525 | 10.5421 | 2.5652 | 20 |

IHS | 13.2541 | 8.3652 | 1.8523 | 40 | |

EHS | 6.5626 | 4.1202 | 1.2301 | 50 | |

Algorithm in this paper | 0.1236 | 0.0952 | 0.0463 | 100 |

Table 2 shows the comparison result of three algorithms, including the best fitness value, the average value of adaptation and optimization of success. Best fitness and average fitness value reflects the convergence rate. The average fitness value reflects the robustness of the algorithm. Success rate reflects the global optimization algorithm. As can be seen from Table 2, the best fitness, success rate, and other performance is better than other algorithms.

In order to analyze the complexity of several algorithms include the standard HS algorithm, IHS algorithm, EHS algorithm, and the proposed algorithm in this paper, the simulation experiment is performed out. The number of iterations is set to 5000. Four algorithms are running 20 times independently. Table 3 shows the average running time (in seconds) of four functions in the same computer (CPU is Intel E6550 2.33 GHz, memory is 2 GB, Windows 7 operating system). The simulation software is MATLAB 2010b. The simulation result shows that the running time of the improved harmony search algorithm in this paper is slightly longer than the standard HS algorithm, lesser than EHS algorithm and close to IHS algorithm. The results show that the algorithm in this paper has better optimization effect without increasing the computational complexity.

Table 3.

Function | HS | IHS | EHS | Algorithm in this paper |

f _{ 1 } | 0.3262 | 0.3346 | 0.4462 | 0.3348 |

f _{ 2 } | 0.4242 | 0.4312 | 0.5402 | 0.4332 |

f _{ 3 } | 0.9738 | 1.0254 | 1.0858 | 1.0254 |

f _{ 4 } | 0.3182 | 0.3313 | 0.4342 | 0.3321 |

Pressure vessel design optimization problem is well known in the field of engineering benchmark problem [ 25 ]. It can be used to optimize the performance test algorithm. As showed in Fig. 8, the optimization problem can be described as to calculate optimized design variables [TeX:] $$x _ { 1 } ( R ) , x _ { 2 } ( L ) , x _ { 3 } \left( t _ { s } \right) and\ x_4(t_h)$$ , which make vessel material most provinces.

Where, * x _{ 1 } * is the radius of the vessel,

Search variables * X(x _{ 1 } ,x _{ 2 } ,x _{ 3 } ,x _{ 4 } ) * , make

Constraint conditions:

[TeX:] $$g _ { 4 } ( X ) = \frac { 1296000 - \frac { 4 } { 3 } \pi x _ { 1 } ^ { 3 } } { \pi x _ { 1 } ^ { 2 } x _ { 2 } } - 1 \leq 0.$$

HS standard algorithms herein, IHS algorithm, EHS algorithm and the improved algorithm with the above parameter HS same 4.1. Fig. 9 shows the convergence curve four algorithms. As can be seen from the figure, the algorithm converges faster and with better fitness than other algorithms. Algorithm runs 20 times. Table 4 shows the results of several optimization algorithms, to adapt the average value of the pressure vessel 6356.17, which is better than the other algorithms results. This article about IHS algorithm is proven to be effective.

The fitness curve and optimization results can explain the simulation results correspond to the above deductions in theory. Firstly, the fitness curves in Figs. 4–7 and Fig. 9 show that the proposed IHS algorithm has faster convergence speed and better optimization results. Next, Table 2 shows the results of four kinds of optimization of complex functions. In Table 2, the average value and the best fitness value reflects the degree fitting algorithm convergence speed. The average fitness value reflects the robustness of the algorithm. Success rate reflects the global optimization algorithm. Table 4 shows that the proposed algorithm can get better than several other harmony search algorithm results. Finally, Table 3 shows the calculation time of each algorithm. The simulation result shows that the running time of the improved harmony search algorithm in this paper is slightly longer than the standard HS algorithm, lesser than EHS algorithm and close to IHS algorithm.

Table 4.

The optimized value | HS | IHS | EHS | The algorithm in this paper |

x _{ 1 } | 48.12 | 54.23 | 40.33 | 44.02 |

x _{ 2 } | 119.82 | 45.36 | 198.23 | 164.24 |

x _{ 3 } | 1.1250 | 1.1250 | 0.8386 | 0.8527 |

x _{ 4 } | 0.7852 | 0.7523 | 0.6052 | 0.4371 |

g _{ 1 } (X) | -0.1744 | -0.0697 | -0.0728 | -0.0037 |

g _{ 2 } (X) | -0.4154 | -0.3123 | -0.3642 | -0.0392 |

g _{ 3 } (X) | -0.5008 | -0.8110 | -0.1741 | -0.3157 |

g _{ 4 } (X) | -0.0485 | -0.4996 | -0.0088 | -0.0605 |

f(X) | 8958.46 | 7199.81 | 6927.12 | 6356.17 |

In practice, many engineering optimization problems belong to the function optimization problem, usually with large-scale, high-dimensional and non-linear characteristics. When a precise optimization algorithms to solve the problem, there is a long computing time disadvantages. Using intelligent optimization algorithm function optimization is an effective method. Therefore, the intelligent optimization algorithm to function optimization problem has important theoretical and practical significance.

The HS algorithm is a new intelligent optimization algorithm. It has a conceptually simple, easy to implement and less adjustment parameters advantages. However, it also has the disadvantage of randomness, such as search directional uncertainty and so on. Although the introduction of different ideas and methods in HS algorithm, but the performance of the algorithm has been improved, and improves the convergence precision and convergence speed of the algorithm. However, HS algorithm and its improved algorithm still has a slower convergence speed, easy to fall into local optimum. At the same time, too many of these parameters IHS algorithm, need to be adjusted through a large number of simulation experiments or experience setting to reduce the applicability of the algorithm in practice. In order to improve the performance of HS algorithm, this paper presents an IHS algorithm. This algorithm optimizes the three important parameters. Based on numerical experiments to optimize complex functions and four pressure vessel problems show that the proposed IHS algorithm is simple, easy to implement, and to find a better solution than the other algorithms more efficiently. The main contribution of this paper includes:

1. These IHS algorithms are only limited to the single parameter optimization. They can’t achieve the overall performance enhancement. Three important parameters are optimized simultaneously in proposed algorithm.

2. An improved new solution generating method refers to the genetic algorithm, which avoid randomness in the process of new solutions generation.

3. In addition, when the size increases, IHS algorithm proposed an overwhelming advantage compared with other HS algorithm.

In general, it can be concluded that the IHS algorithm, its simple, high-quality solutions to achieve, few set parameters and faster convergence. In dealing with other complex optimization algorithm is an ideal method. IHS algorithm can be used to optimize some difficulties and problems, multidimensional better choice in the real world.

He received the Ph.D. degree in control theory and control engineering from Northeastern University, China in 2013. His research interests include predictive control, delay compensation and scheduling for networked control system and optimization algorithms. He is currently a lecturer at Shenyang University of Technology, Shenyang, China.

He received the B.E. degree in electrical and information engineering from Zaozhuang University, China, in 2014. He is currently pursuing his M.E. degree in control engineering at Shenyang University of Technology, Shenyang, China. His current research interests include networked control system and optimization algorithms.

- 1 Z. W . Geem, J. H. Kim, G. V . Loganathan, "A new heuristic optimization algorithm: harmony search,"
*Simulations, 2001*, vol. 76, no. 2, pp. 60-68. doi:[[[10.1177/003754970107600201]]] - 2 H. B. Ouyang, L. Q. Gao, D. X. Zou, X. Y . Kong, "Exploration ability study of harmony search algorithm and its modification,"
*Control Theory and Applications, 2014*, vol. 31, no. 1, pp. 57-65. custom:[[[-]]] - 3 D. Manjarres, I. Landa-Torres, S. Gil-Lopez, J. D. Ser, M. N. Bilbao, S. Salcedo-Sanz, Z. W. Geem, "A survey on applications of the harmony search algorithm,"
*Engineering Applications of Artificial Intelligence, 2013*, vol. 26, no. 8, pp. 1818-1831. doi:[[[10.1016/j.engappai.2013.05.008]]] - 4 B. Alatas, "Chaotic harmony search algorithms,"
*Applied Mathematics and Computation, 2010*, vol. 216, no. 9, pp. 2687-2699. doi:[[[10.1016/j.amc.2010.03.114]]] - 5 R. Arul, G. Ravi, S. Velusami, "Solving optimal power flow problems using chaotic self-adaptive differential harmony search algorithm,"
*Electric Power Components and Systems, 2013*, vol. 41, no. 8, pp. 782-805. doi:[[[10.1080/15325008.2013.769033]]] - 6 S. Sayah, A. Hamouda, A. Bekra, "Efficient hybrid optimization approach for emission constrained economic dispatch with nonsmooth cost curves,"
*International Journal of Electrical Power and Energy Systems, 2014*, vol. 56, pp. 127-139. doi:[[[10.1016/j.ijepes.2013.11.001]]] - 7 B. Zeng, Y. Dong, "An improved harmony search based energy-efficient routing algorithm for wireless sensor networks,"
*Applied Soft Computing, 2016*, vol. 41, pp. 135-147. doi:[[[10.1016/j.asoc.2015.12.028]]] - 8 G. F. de Medeiros, M. Kripka, "Optimization of reinforced concrete columns according to different environmental impact assessment parameters,"
*Engineering Structures, 2014*, vol. 59, pp. 185-194. doi:[[[10.1016/j.engstruct.2013.10.045]]] - 9 X. Y. Li, K. Qin, B. Zeng, L. Gao, J. Z. Su, "Assembly sequence planning based on an improved harmony search algorithm,"
*International Journal of Advanced Manufacturing Technology, 2016*, vol. 84, no. 9, pp. 2367-2380. doi:[[[10.1007/s00170-015-7873-9]]] - 10 G. Naresh, M. R. Raju, S. V . L. Narasimham, "Coordinated design of power system stabilizers and TCSC employing improved harmony search algorithm,"
*Swarm and Evolutionary Computation, 2016*, vol. 27, pp. 169-179. doi:[[[10.1016/j.swevo.2015.11.003]]] - 11 Z. D. Tian, S. J. Li, Y . H. W ang, X. D. W ang, "LSSVM predictive control for calcination zone temperature in rotary kiln with IHS algorithm,"
*Journal of Harbin Institute of Technology (New Series), 2016*, vol. 23, no. 4, pp. 67-74. custom:[[[-]]] - 12 M. Mahdavi, M. Fesanghary, E. Damangir, "An improved harmony search algorithm for solving optimization problems,"
*Applied Mathematics and Computation, 2007*, vol. 188, no. 2, pp. 1567-1579. doi:[[[10.1016/j.amc.2006.11.033]]] - 13 M. G. H. Omran, M. Mahdavi, "Global-best harmony search,"
*Applied Mathematics and Computation, 2008*, vol. 198, no. 2, pp. 643-656. doi:[[[10.1016/j.amc.2007.09.004]]] - 14 M. Fesangharya, M. Mahdavib, M. Minary-Jolandan, Y. Alizadeha, "Hybridizing harmony search algorithm with sequential quadratic programming for engineering optimization problems,"
*Computer Methods in Applied Mechanics and Engineering, 2008*, vol. 197, no. 33-40, pp. 3080-3091. doi:[[[10.1016/j.cma.2008.02.006]]] - 15 S. Das, A. Mukhopadhyay, A. Roy, A. Abraham, B. K. Panigrahi, "Exploratory power of the harmony search algorithm: analysis and improvements for global numerical optimization,"
*IEEE Transactions on SystemsMan, and Cybernetics, Part B (Cybernetics), , 2011*, vol. 41, no. 1, pp. 89-106. doi:[[[10.1109/TSMCB.2010.2046035]]] - 16 E. Valian, S. Tavakoli, S. Mohanna, "An intelligent global harmony search approach to continuous optimization problems,"
*Applied Mathematics and Computation, 2014*, vol. 232, no. 3, pp. 670-684. doi:[[[10.1016/j.amc.2014.01.086]]] - 17 D. X. Zou, L. Q. Gao, J. H. Wu, S. Li, "Novel global harmony search algorithm for unconstrained problems,"
*Neurocomputing, 2010*, vol. 73, no. 16, pp. 3308-3318. doi:[[[10.1016/j.neucom.2010.07.010]]] - 18 W. L. Xiang, M. Q. An, Y. Z. Li, R. C. He, J. F. Zhang, "An improved global-best harmony search algorithm for faster optimization,"
*Expert Systems with Applications, 2014*, vol. 41, no. 13, pp. 5788-5803. doi:[[[10.1016/j.eswa.2014.03.016]]] - 19 Z. W . Geem, "Effects of initial memory and identical harmony in global optimization using harmony search algorithm,"
*Applied Mathematics and Computation, 2012*, vol. 218, no. 22, pp. 11337-11343. doi:[[[10.1016/j.amc.2012.04.070]]] - 20 Z. W. Geem, K. B. Sim, "Parameter-setting-free harmony search algorithm,"
*Applied Mathematics and Computation, 2010*, vol. 217, no. 8, pp. 3881-3889. doi:[[[10.1016/j.amc.2010.09.049]]] - 21 M. Khalili, R. Kharrat, K. Salahshoor, M. H. Sefat, "Global dynamic harmony search algorithm: GDHS,"
*Applied Mathematics and Computation, 2014*, vol. 228, pp. 195-219. doi:[[[10.1016/j.amc.2013.11.058]]] - 22 H. B. Ouyang, L. Q. Gao, S. LI, X. Y . Kong, Q. Wang, D. X. Zou, "Improved harmony search algorithm: LHS,"
*Applied Soft Computing Journal, 2017*, vol. 53, pp. 133-167. doi:[[[10.1016/j.asoc.2016.12.042]]] - 23 T. Hassanzadeh, H. R. Kanan,
*"Fuzzy FA: a modified firefly algorithm,: Applied Artificial Intelligence, 2014*, vol. 28, no. 1, pp. 47-65. custom:[[[-]]] - 24 G. Z. Tan, K. Bao, R. M. Rimiru, "A composite particle swarm algorithm for global optimization of multimodal functions,"
*Journal of Central South University, 2014*, vol. 21, no. 5, pp. 1871-1880. doi:[[[10.1007/s11771-014-2133-y]]] - 25 J. Kruzelecki, R. Proszowski, "Shape optimization of thin-walled pressure vessel end closures,"
*Structural and Multidisciplinary Optimization, 2012*, vol. 46, no. 5, pp. 739-754. doi:[[[10.1007/s00158-012-0789-1]]]